The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.
Linear maps preserving the Cullis' determinant. II
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
This paper is the second in the series of papers devoted to the explicit description of linear maps preserving the Cullis' determinant of rectangular matrices with entries belonging to an arbitrary ground field which is large enough. In this part we solve the linear preserver problem for the Cullis' determinant defined on the spaces of matrices of size $n\times k$ with $k \ge 4,\; n \ge k + 2$ and $n + k$ is odd. In comparison with the case when $n + k$ is even, in this case linear maps preserving the Cullis' determinant could be singular and are represented as a sum of two linear maps: first is two-sided matrix multiplication and second is any linear map whose image consists of matrices, all rows of which are equal.
fields
math.CO 2years
2026 2verdicts
UNVERDICTED 2representative citing papers
All maps phi and psi satisfying the polynomial preservation equation P(x + λ y) = P(phi(x) + λ psi(y)) are explicitly described using the gradient span L_P for homogeneous P over fields of characteristic zero (and under extra conditions in positive characteristic).
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The Cullis' determinant as Pfaffian
The Cullis' determinant of rectangular matrix X equals the Pfaffian of a matrix obtained from X by multiplication and transposition, enabling an efficient polynomial-time algorithm.
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Nonlinear maps preserving the polynomial
All maps phi and psi satisfying the polynomial preservation equation P(x + λ y) = P(phi(x) + λ psi(y)) are explicitly described using the gradient span L_P for homogeneous P over fields of characteristic zero (and under extra conditions in positive characteristic).